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Perspective Monocular and Binocular Vision 22

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PERSPECTIVE : MONOCULAR AND BINOCULAR VISION 22. Geometrical Perspective. The perspective of an object or of a group of objects (from the Latin : to see across) is the trace of all the points of intersection of all the straight lines from a fixed point (the viewpoint or centre of projection) to all the points of the objects to be represented, with a certain surface called the surface of projection. This surface is generally a vertical plane, but is sometimes a cylindrical surface (panoramas), or a segment of a sphere (cupolas), or, more rarely, some other surface.

Practically, according to Leonardo da Vinci, the perspective may be defined as the trace which would be obtained on a transparent surface (glass, or gauze stretched on a frame in the case of a plane perspective), when one eye is kept in a fixed position determined by a sighthole, and the other is closed, in such a way that each of the points or outlines of this trace exactly masks the point or the corresponding outline in the subject to be represented.

The perspective of anything of which all the parts, whether real or imaginary, have known dimensions and occupy known positions can be obtained by relatively simple geometrical constructions. Conversely, if a perspective con tains the images of certain known objects, it is possible to deduce from it the dimensions and the relative positions of other unknown objects whose images figure in that perspective.

Such a perspective regarded by one eye only from exactly the position of the viewpoint would appear to us, at least as far as the forms are concerned (without considering colours and luminosities), just like the object represented would appear when viewed from the correspond ing point, the same outlines being seen in the same relative positions.

In conformity with this definition, the surface of a projection plane only plays the role of an open window through which appears the land scape or the scene which was represented.

23. If we consider an object (Fig. 4) which for clearness has been purposely chosen of simple form, a viewpoint 0, and a vertical plane then the perpendicular OP dropped to the plane from the viewpoint meets the plane at a point P (called the principal point), the distance OP being the principal distance of the perspective obtained.

Any group of straight lines parallel to one another and to the plane of projection will be reproduced in the perspective by straight lines parallel to those considered. In particular, all vertical lines in the subject will be represented by vertical lines in the perspective.

Any groups of parallel straight lines which are not parallel to the projection plane will be represented in the perspective by a group of straight lines converging to the same vanishing point, which is defined by the intersection of the projection plane with a straight line dropped from the viewpoint parallel to the direction in question in the subject.

The vanishing points of all the horizontal lines are situated on the principal horizontal JIM', the intersection of the projection plane with the horizontal plane through the viewpoint and also (in this case of a vertical projection plane) through the principal point P.

In particular, all the horizontals contained in the facade of the shed (Fig. 4), or parallel to this facade, are represented by straight lines which converge to the vanishing point F defined by the intersection of the plane of projec tion with the straight line OF dropped from the viewpoint parallel to the straight lines being considered in the subject. All other groups of lines parallel to the facade of the shed will have their vanishing points on the vertical line FG.

24. Once the position of the viewpoint and the direction of the projection plane have been determined, the perspective obtained is to a. close degree independent of the principal dis tance. The perspectives obtained from a single viewpoint but on several parallel planes are geometrically similar ; any one can be changed into any other merely by proportional ampli fication or reduction ; for example, by means of a pantograph. The principal distance only affects the scale of the images, which all vary proportionally.

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